The invention relates generally to a numerical simulation. More particularly, the invention relates to a method for automatically decomposing a surface or volume to generate from a geometric model the finite elements to be used in the simulation. The process has application for structural analysis (including linear elasticity, vibration analysis and plasticity), electric and magnetic field analysis, fluid mechanics, heat transfer, solid modelling and various other applications. While the invention will be described in terms of decomposing a volume or surface it will be understood that the same techniques may be applied to other problem solutions.
Analytical solutions have been the historically dominant approach to many problems of this type. Analytical solutions are mathematically logical representations of problems that can also be solved with a numerical solution.
Despite the wide use of analytical solutions, numerical solutions such as the finite element method have become increasingly popular in the wake of advancing computer technology and is an accepted method of analysis both in industry and academia. The invention uses the finite element method. This method or process solves problems using convergent numerical analysis. The method mathematically defines a plurality of finite element which are representations of infinitesimal sections of a continuum.
An advantage of the finite element method over analytical solutions is that the finite element method can be applied to a much larger range of problems and is not limited to a small class of solution types as is the analytical solutions approach.
The finite element method includes three sequential steps: preprocessing, analysis, and post processing. The invention is concerned with the preprocessing phase in which a surface or volume is decomposed into finite elements. The analysis phase utilizes numerical methods and considerable emphasis has been placed on its development in the prior art. The existing techniques are quite sophisticated. The post processing phase involves the handling of large amounts of data produced by the analysis phase in order to present that data in a comprehensible manner. Both the analysis and post processing phases have been developed to a high level of sophistication. Commercial analysis packages exist to do analysis in specific technical areas such as linear elasticity, plasticity, heat transfer and fluids analysis.
The prior art includes various methods described in the following U.S. Patents. U.S. Pat. No. 4,941,114 describes a triangular mesh generation. It is generally recognized that finite elements that approach a cube of square are a more precise basis for analysis. Thus, the approach described in the patent is not a preferred approach to solving such problems.
U.S. Pat. No. 4,797,842 attempts to define a series of locus points that are used to define a natural flow or natural breakup of an object would naturally suggest a unique mathematical process. This approach is seemingly limited to quadrilaterals and triangles in the mesh generation process. Thus, inherently the approach in this patent also uses hexahedrons and prisms for volumes, quadrilaterals and triangles for surfaces. The accuracy of this approach is unsatisfactory because of the use of triangles in surface decomposition and prisms in surface decomposition. In addition the use of locus points requires additional data processing that is not necessary with the present invention.
U.S. Pat. No. 4,183,013 describes a system for extracting shape features from an object. The teachings are strictly limited to two dimensional surfaces and do not disclose a method for generating a finite element mesh.
It is generally recognized that a decomposition process that defines quadrilaterals or hexahedrons is preferable because the results are more precise. The prior art computerized surface and volume decomposition procedures have not been satisfactory because they cannot decompose many surfaces and volumes into respectively only quadrilaterals or hexahedrons.
Still another problem is that the prior art surface decomposition techniques do not lead to results which are as accurate as the method in accordance with the present invention. In some cases the prior art techniques have not conceptualized the procedures for decomposing all of the many shapes that are possible in an object.
It is an object of the invention to provide a more accurate and complete decomposition of surfaces and volumes.
It is an object of the invention to provide finite element generation that defines elements that approach either a square or cube shape.